Transient phenomena in a random walk

The paper studies the limit distributions of the maximum of sums $\max_{1\le k\le n}\sum_{l=1}^k\xi_{n,l}$ for the triangular array $\xi_{n,k}$, $k=1,\ldots,n$, $n=1,2,\ldots $, of independent identically distributed random variables in a singular series in cases where $a_n=E\xi_{n,k}\to 0$ and/or 1) $a_n\sqrt n\to\infty$, or 2) $a_n\sqrt n\to-\infty$, or 3) $a_n\sqrt n\to 0$ as $n\to\infty$. The direct proof that the analytic expressions for limit laws coincide was previously obtained by different authors and is given. Moreover, for these transient cases the convergence of the sequence of distributions of maximums to the limit laws is proved with the help of the characteristic functions method.